IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v123y2013i8p3239-3272.html
   My bibliography  Save this article

Stochastic Burgers PDEs with random coefficients and a generalization of the Cole–Hopf transformation

Author

Listed:
  • Englezos, Nikolaos
  • Frangos, Nikolaos E.
  • Kartala, Xanthi-Isidora
  • Yannacopoulos, Athanasios N.

Abstract

This paper studies forward and backward versions of the random Burgers equation (RBE) with stochastic coefficients. First, the celebrated Cole–Hopf transformation reduces the forward RBE to a forward random heat equation (RHE) that can be treated pathwise. Next we provide a connection between the backward Burgers equation and a system of FBSDEs. Exploiting this connection, we derive a generalization of the Cole–Hopf transformation which links the backward RBE with the backward RHE and investigate the range of its applicability. Stochastic Feynman–Kac representations for the solutions are provided. Explicit solutions are constructed and applications in stochastic control and mathematical finance are discussed.

Suggested Citation

  • Englezos, Nikolaos & Frangos, Nikolaos E. & Kartala, Xanthi-Isidora & Yannacopoulos, Athanasios N., 2013. "Stochastic Burgers PDEs with random coefficients and a generalization of the Cole–Hopf transformation," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3239-3272.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:8:p:3239-3272
    DOI: 10.1016/j.spa.2013.03.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414913000653
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2013.03.001?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Bonnet, Guillaume & Adler, Robert J., 2007. "The Burgers superprocess," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 143-164, February.
    2. Hodges, Stewart & Carverhill, Andrew, 1993. "Quasi Mean Reversion in an Efficient Stock Market: The Characterisation of Economic Equilibria which Support Black-Scholes Option Pricing," Economic Journal, Royal Economic Society, vol. 103(417), pages 395-405, March.
    3. Ma, Jin & Yong, Jiongmin, 1997. "Adapted solution of a degenerate backward spde, with applications," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 59-84, October.
    4. Yannacopoulos, Athanasios N., 2008. "Rational expectations models: An approach using forward-backward stochastic differential equations," Journal of Mathematical Economics, Elsevier, vol. 44(3-4), pages 251-276, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Xanthi-Isidora Kartala & Nikolaos Englezos & Athanasios N. Yannacopoulos, 2020. "Future Expectations Modeling, Random Coefficient Forward–Backward Stochastic Differential Equations, and Stochastic Viscosity Solutions," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 403-433, May.
    2. Lüders, Erik & Peisl, Bernhard, 2001. "How do investors' expectations drive asset prices?," ZEW Discussion Papers 01-15, ZEW - Leibniz Centre for European Economic Research.
    3. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    4. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    5. Confortola, Fulvia, 2007. "Dissipative backward stochastic differential equations with locally Lipschitz nonlinearity," Stochastic Processes and their Applications, Elsevier, vol. 117(5), pages 613-628, May.
    6. Ma, Jin & Yin, Hong & Zhang, Jianfeng, 2012. "On non-Markovian forward–backward SDEs and backward stochastic PDEs," Stochastic Processes and their Applications, Elsevier, vol. 122(12), pages 3980-4004.
    7. Leonenko, N. N. & Woyczynski, W. A., 1998. "Exact parabolic asymptotics for singular -D Burgers' random fields: Gaussian approximation," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 141-165, August.
    8. Mirosław Lachowicz & Henryk Leszczyński, 2020. "Modeling Asymmetric Interactions in Economy," Mathematics, MDPI, vol. 8(4), pages 1-14, April.
    9. Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2023. "Spatial growth theory: Optimality and spatial heterogeneity," Journal of Economic Dynamics and Control, Elsevier, vol. 146(C).
    10. Yin, Hong, 2014. "Solvability of forward–backward stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2583-2604.
    11. Du, Kai & Meng, Qingxin, 2010. "A revisit to -theory of super-parabolic backward stochastic partial differential equations in," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1996-2015, September.
    12. Sundar, P. & Yin, Hong, 2009. "Existence and uniqueness of solutions to the backward 2D stochastic Navier-Stokes equations," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1216-1234, April.
    13. McDonald, Stuart, 2006. "Finite Difference Approximation for Linear Stochastic Partial Differential Equations with Method of Lines," MPRA Paper 3983, University Library of Munich, Germany, revised 30 May 2007.
    14. Jiang-Lun Wu & Wei Yang, 2013. "A Galerkin approximation scheme for the mean correction in a mean-reversion stochastic differential equation," Papers 1305.1868, arXiv.org.
    15. Lüders, Erik, 2002. "Asset Prices and Alternative Characterizations of the Pricing Kernel," ZEW Discussion Papers 02-10, ZEW - Leibniz Centre for European Economic Research.
    16. Weinbaum, David, 2009. "Investor heterogeneity, asset pricing and volatility dynamics," Journal of Economic Dynamics and Control, Elsevier, vol. 33(7), pages 1379-1397, July.
    17. Qiu, Jinniao, 2017. "Weak solution for a class of fully nonlinear stochastic Hamilton–Jacobi–Bellman equations," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1926-1959.
    18. Phoebe Koundouri & Georgios I. Papayiannis & Athanasios Yannacopoulos, 2022. "Optimal Control Approaches to Sustainability under Uncertainty," DEOS Working Papers 2215, Athens University of Economics and Business.
    19. Du, Kai & Zhang, Qi, 2013. "Semi-linear degenerate backward stochastic partial differential equations and associated forward–backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1616-1637.
    20. Zura Kakushadze, 2014. "Mean-Reversion and Optimization," Papers 1408.2217, arXiv.org, revised Feb 2016.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:123:y:2013:i:8:p:3239-3272. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.